I have recently been pondering the concept of the irrational number. This is a number which cannot be expressed as the ratio of two integers. The Pythagoreans rejected this concept and, according to legend, executed a member who asserted (and proved) its existence. An example is the square root of 2 - the diagonal / side ratio of any square. A little thought leads to the following logical implication of the irrational number concept (quoting a Wikipedia author):
" When the ratio of lengths of two line segments is an irrational number, the line segments are also described as being incommensurable, meaning that they share no "measure" in common, that is, there is no length ("the measure"), no matter how short, that could be used to express the lengths of both of the two given segments as integer multiples of itself."
To me, this is a weirdity that rivals (perhaps exceeds) the weirdities of Quantum Mechanics. Quantum weirdities are concerned with the weird behavior of material objects. The irrational number is a weirdity of CONCEPT - apart from any problems of measurement, construction or material existence. I simply cannot form a valid and self consistent CONCEPT of such a quantity.
Somebody please straighten me out!
Bob Sciamanda
Physics, Edinboro Univ of PA (Em)
treborsciamanda@gmail.com
www.sciamanda.com